Division of Molecular Biosciences, Theoretical Systems Biology Group, Imperial College London, London SW7 2AZ, United Kingdom.
J Chem Phys. 2013 May 7;138(17):174101. doi: 10.1063/1.4802475.
Moment approximation methods are gaining increasing attention for their use in the approximation of the stochastic kinetics of chemical reaction systems. In this paper we derive a general moment expansion method for any type of propensities and which allows expansion up to any number of moments. For some chemical reaction systems, more than two moments are necessary to describe the dynamic properties of the system, which the linear noise approximation is unable to provide. Moreover, also for systems for which the mean does not have a strong dependence on higher order moments, moment approximation methods give information about higher order moments of the underlying probability distribution. We demonstrate the method using a dimerisation reaction, Michaelis-Menten kinetics and a model of an oscillating p53 system. We show that for the dimerisation reaction and Michaelis-Menten enzyme kinetics system higher order moments have limited influence on the estimation of the mean, while for the p53 system, the solution for the mean can require several moments to converge to the average obtained from many stochastic simulations. We also find that agreement between lower order moments does not guarantee that higher moments will agree. Compared to stochastic simulations, our approach is numerically highly efficient at capturing the behaviour of stochastic systems in terms of the average and higher moments, and we provide expressions for the computational cost for different system sizes and orders of approximation. We show how the moment expansion method can be employed to efficiently quantify parameter sensitivity. Finally we investigate the effects of using too few moments on parameter estimation, and provide guidance on how to estimate if the distribution can be accurately approximated using only a few moments.
矩近似方法因其在化学反应系统随机动力学近似中的应用而受到越来越多的关注。在本文中，我们为任何类型的倾向性推导出了一种通用的矩展开方法，该方法允许展开到任意数量的矩。对于一些化学反应系统，需要超过两个矩来描述系统的动态特性，而线性噪声近似无法提供这些特性。此外，即使对于均值对高阶矩没有强烈依赖的系统，矩近似方法也可以提供关于基础概率分布的高阶矩的信息。我们使用二聚化反应、米氏动力学和 p53 系统的模型来演示该方法。我们表明，对于二聚化反应和米氏酶动力学系统，高阶矩对均值的估计影响有限，而对于 p53 系统，均值的解需要几个矩才能收敛到从许多随机模拟中获得的平均值。我们还发现，低阶矩的一致性并不能保证高阶矩的一致性。与随机模拟相比，我们的方法在捕捉随机系统的行为方面具有很高的数值效率，无论是在平均值还是高阶矩方面，并且我们还提供了不同系统大小和逼近阶数的计算成本表达式。我们展示了如何使用矩展开方法有效地量化参数敏感性。最后，我们研究了使用太少矩对参数估计的影响，并提供了如何估计仅使用几个矩是否可以准确近似分布的指导。
